Bring a large pot of salted water to a boil. Cook macaroni according to package directions until al dente. Reserve ¼ cup pasta water, then drain.
In a large skillet or saucepan over medium heat, melt butter. Whisk in flour and cook for 1 minute, stirring constantly.
Slowly whisk in milk and cook until slightly thickened, about 2–3 minutes.
Reduce heat to low. Stir in cheddar, mozzarella, and Parmesan until melted and smooth.
Add salt, black pepper, and paprika if using. Stir to combine.
Add cooked chicken and drained macaroni to the cheese sauce. Toss to coat evenly. Use reserved pasta water if needed to loosen the sauce.
Remove from heat and serve warm.